THE PROBLEM OF PHASE DIAGRAMS OF COMPLEX AQUEOUS SYSTEMS VARYING CONCENTRATION AND TEMPERATURE

The work of J. Willard Gibbs at the turn from 19th to 20th century revolutionized Thermodynamics, defining Chemical Thermodynamics and Statistical Mechanics. Very mathematical, but integrated in the beginning of Physical Chemistry, due to his Ph.D. as chemical engineer at Yale, together with his extremely broad intellectual interests and a 3 years stay in Europe, with connections with chemists and physicists.
Some basic advances in the following decades must be mentioned for the understanding of the problem treated here. The work of Peter Debye (Nobel in Chemistry in 1936) in the Debye – Huckel (DH) theory of electro-chemical equilibrium was continued by Lars Onsager (Nobel in Chemistry in 1968). Onsager also studied chemical engineering in Norway, but his inclinations were mainly intellectual, with some difficulties in human communication. His interest in electrolytes was permanent and he developed a deep appreciation of the relation of theory to experiment. His achievements included correction in DH theory, Reciprocal Relations in Irreversible Processes, solution of Ising model in 2 D, but also phase transition of cylinders with concentration, effects of shape on the interactions of colloidal particles, thermal diffusion and turbulence. Onsager Nobel lecture was on “The motion of ions: principles and concepts”, mentioning at the end H-Bond chains and electrically active defects in water in connection to problems in biology.
In parallel the work of Lev Landau (Nobel of Physics in 1962) in the 30’s included second order phase transitions but also the DLVO theory of colloidal stability on interactions between colloidal particles and their aggregation behavior.

Here the focus is on Lyotropic Liquid Crystals (LLC), a subject of Physical Chemistry. From the millennial practice of soap production, the chemical industry of soaps developed in the 19th century, under the principle that soap cleans because amphiphile molecules bind to fat, not soluble in water. Lyotropic mixtures (amphiphiles / water / additives) started to be scientifically studied by chemists in the beginning of the 20th century and already in 1913 the concept of “micelle” (aggregate of amphiphile molecules) was introduced. In the 30’s the two basic structures of LLC were already known from X-ray diffraction: lamellar neat phase and hexagonal middle phase.
Ternary phase diagrams of amphiphiles / solvents / additives, varying concentration at fixed temperature, were systematically studied since the 60’s by the Swedish group of physical chemists in Lund and by the Luzzati group in Strasbourg, who focused also the order / disorder transition of hydrocarbon chains, since LLC existed only above this transition. And it was soon discovered that biological membranes also presented such melting transition, defining the field of bio-mimetic membranes.
The theory of micelle formation was proposed by Tanford in 1974, and the geometrical packing parameter p was introduced by Israelachvili in 1976, unifying micelles and bilayers. The field of thermotropic liquid crystals (phase transitions with temperature) crossed with LLC with the discovery of nematic lyotropic phases by chemists varying concentration, compound and counter ions, until the two nematic phases of opposed symmetry were discovered in 1976 by Reeves (a chemist) using NMR. Physicists then entered in the field, varying also temperature.
I proposed the field of LLC at IFUSP in 1974, after a short interaction with Reeves. In my Frei Dozent Thesis (1982) several experimental techniques were used, but I also made analysis of the origin of magnetic orientation and used DLVO theory for inter-micellar interactions. Over the next decades I engaged in several different subjects, with emphasis on the changes of the transient micellar object across phase transitions, and I mention some theoretical works:
– The characteristics of cylinders in hexagonal phases were shown to correlate with changes in size and flexibility across several different phase transitions (from isotropic and nearby cubic phases) varying concentration by simple analysis of the exponent of X-ray data with volume concentration [1].
– The change of micellar object from prolate ellipsoid to spherocylinder with concentration at the iso-hexagonal transition was explained in terms of elastic bending energy [2].
– The change of micellar object from spherocylinder to square tablet, which occurs geometrically in a continuous way, with an intermediate biaxial object, was calculated in terms of micellar elastic bending energy in the surfactant parameter model, and is able to explain the nematic cylindrical – nematic discotic phase transitions in three different LLC systems [3].
– The phase diagram for the transition between the two nematic symmetries in LLC, in function of variations in concentration and temperature, may occur either with a biaxial intermediate phase, or directly, depending on the specific chemicals of the samples. It is proposed that this may be related to changes of uniaxial micellar form, which may occur either smoothly (biaxial intermediate phase) or abruptly (directly). This was the first statistical microscopic approach able to model experimentally observed lyotropic biaxial nematic phases [4].

After this last publication [4] I have been invited to present a review paper on micelles [5] and our work inspired theoretical works of other scientists [6].
I have worked also for two decades with bio-mimetic membranes, but such work is outside the scope of the present event.

References
[1] “Micellar Growth in Hexagonal Phases of Lipid Systems”, P. Mariani and L.Q. Amaral, Physical Review E 50, 1678-1681 (1994).
[2] “Bending Energy and the relative stability of Micellar Forms”, G. Taddei and L.Q. Amaral, Journal of Physical Chemistry 96, 6102-6104 (1992).
[3] “Change in Micelle Form Induced by Cosurfactant Addition in Nematic Lyotropic Phases”, L.Q.Amaral, O.Santin Filho, G.Taddei and N.Vila-Romeu, Langmuir 13, 5016-5021 (1997).
[4] “Mixture of changing uniaxial micellar forms in lyotropic biaxial nematics”, Henriques EF, Passos CB, Henriques VB, Amaral LQ, Liquid Crystals 35 (5), 555-568 (2008).
[5] “Micelles forming biaxial lyotropic nematic phases”, L. Q. Amaral, Liquid Crystals 37 (6), 627 – 640 (2010).
[6] “New phase diagrams in the mixture of rods and plates of biaxial nematic liquid crystals”, Mukherjee PK, Journal of Molecular Liquids 220: 742–746 (2016).

The imaginary world of free parafermions

The Baxter-Fendley model of Z(N) parafermions is a relatively simple N-state generalization of the quantum Ising chain. The energy eigenspectrum of this non-Hermitian model, subject to open boundary conditions, is composed of free parafermions, which are a natural generalization of free fermions to the complex plane. The model has remarkable physical properties, including boundary-dependent bulk behaviour, which is an example of the non-Hermitian skin effect. In this talk, based on work with Alex Henry, I will discuss the appearance of exceptional points in the eigenspectrum.

The quantum Mpemba effect and entanglement asymmetry

Symmetry and symmetry breaking are two pillars of modern quantum physics. However, quantifying how much a symmetry is broken is an issue that has received little attention. In extended quantum systems, this problem is intrinsically bound to the subsystem of interest. In this talk, we borrow methods from the theory of entanglement in many-body quantum systems to introduce a subsystem measure of symmetry breaking that we dub entanglement asymmetry. As a prototypical illustration, we study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global U(1) symmetry is restored dynamically. We find the counterintuitive result that more the symmetry is initially broken, faster it is restored, a quantum Mpemba effect.

Is History foreseeable? Revealing hidden structures of historical events using complex networks

Recently historians have become interested in the use of complex networks (graphs) in historiographical research. Historical events may involve hundreds of actors and span decades or even centuries. Networks are helping shed a new light onto historical mechanisms which have hitherto gone unnoticed. Moreover it has helped bring about a paradigm shift from Thomas Carlyle’s influential “Great Man History” to that of “History from below”. In this talk I will discuss my  interaction with experts on the History of Anglo-Saxon Great Britain and the analysis of Bede’s “Ecclesiastical History of the English People”. Bede (ca. 672 – 735 A.D.) was a monk and is regarded as the father of English historiography. He was also an important scholar who helped preserve and disseminate mathematical and scientific knowledge during the lower Middle Ages. One of the main findings of our results is the surprising role Bede ascribed to women in the early history of Great Britain, confirming the conjectures of modern historiography that they were simply more than weavers of peace and had important roles in the networks they were part of.

A phase transition characterized via chaotic diffusion for two dimensional mappings

In this talk we discuss some characteristics of scaling invariance for a transition from integrability to non-integrability in a class of dynamical systems described by a two-dimensional, nonlinear and area-preserving mapping. The dynamical variables are action $I$ and angle $\theta$ and that the angle diverges in the limit of vanishingly action. The transition is controlled by a parameter $\epsilon$ closely related to the order parameter. A scaling invariance observed for the average
squared action along the chaotic sea gives evidence that the transition observed from integrability to non-integrability is equivalent to a second-order, also called continuous, phase transition since when the order parameter approaches zero at the same time the response of the order parameter to the conjugate field (susceptibility) diverges. This investigation allows application to a wide class of systems and transitions, including transition from limited to unlimited chaotic diffusion in dissipative systems and also in a transition from limited to unlimited Fermi acceleration in time-dependent billiard systems.

Phase transitions in antagonistic binary lattice gases, in and out of equilibrium

Studies of binary fluids with hard-core repulsive interactions between particles of different species have contributed much to the theory of entropically driven phase separation. Widom and Rowlinson showed that in equilibrium, a binary mixture with hard-sphere interactions between opposite species, and no interaction at all between molecules of the same species, phase separates above a critical density. In the Widom-Rowlinson lattice gas (WRLG), two particle species (A, B) diffuse freely via particle-hole exchange, subject to both on-site exclusion and prohibition of A-B nearest-neighbor pairs. As an athermal system, the overall densities are the only control parameters. As the densities increase, an Isinglike phase transition occurs, leading to ordered states with A- and B-rich domains separated by hole-rich interfaces. A version of the WRLG maintained out of equilibrium by a drive that favors particle moves along one direction (the DRWLG) displays atypical collective behavior such as kink singularities in the structure factor, maxima at non-vanishing wavevector values, oscillating correlation functions, and phase separation into multiple striped domains perpendicular to the drive, with a preferred wavelength depending on density and drive intensity. Attempts to understand these behaviors via a continuum description, diagrammatic perturbation theory and renormalization-group approaches are ongoing. Finally, irreversible random sequential adsorption of WRLG particles yields challenging lattice combinatorial problems on the line as well as intriguing percolative transitions on two-dimensional lattices. I shall review the behavior of these models and highlight open questions.

Bethe Ansatz-like solution for a “lifted” Asymmetric Exclusion Process

There has been considerable progress in recent years in improving the performance of Markov-Chain-Monte-Carlo algorithms by employing a tool called “lifting”.
This can be applied to stochastic processes more generally as a means speed up the convergence of associated computational problems. I discuss a lifting of the
Totally Asymmetric Simple Exclusion Process and show that it — surprisingly — can be solved exactly by a Bethe-like Ansatz.

A time-dependent approach to inelastic scattering spectroscopies in and away from equilibrium: beyond perturbation theory

I present a new computational paradigm to simulate time and momentum resolved inelastic scattering spectroscopies in correlated systems. The conventional calculation of scattering cross sections relies on a treatment based on time-dependent perturbation theory, that provides formulation in terms of Green’s functions. In equilibrium, it boils down to evaluating a simple spectral function equivalent to Fermi’s golden rule, which can be solved efficiently by a number of numerical methods. However, away from equilibrium, the resulting expressions require a full knowledge of the excitation spectrum and eigenvectors to account for all the possible allowed transitions, a seemingly unsurmountable complication. Similar problems arise when the quantity of interest originates from higher order processes, such as in Auger, Raman, or resonant inelastic X-ray scattering (RIXS). To circumvent these hurdles, we introduce a time-dependent approach that does not require a full diagonalization of the Hamiltonian: we simulate the full scattering process, including the incident and outgoing particles (neutron, electron, photon) and the interaction terms with the sample, and we solve the time-dependent Schrödinger equation. The spectrum is recovered by measuring the momentum and energy lost by the scattered particles, akin an actual energy-loss experiment. The method can be used to study transient dynamics and spectral signatures of correlation-driven non-equilibrium processes, as I illustrate with several examples and experimental proposals using the time-dependent density matrix renormalization group method as a solver. Even in equilibrium, we find higher order contributions to the spectra that can potentially be detected by modern instruments.

Hidden symmetries

The development of exact and non-perturbative methods for the study of non-linear phenomena presents an unusual aspect. The relevant symmetries behind such non-linear phenomena are not in general symmetries of the Lagrangian (or Hamiltonian) of the system, but hidden symmetries of auxiliary operators that linearize, in some sense, the dynamics of the theory. The crucial concept is that of path independence for some connections that lead to an iso-spectral time evolution of these auxiliary operators. We discuss how such an idea applies to several areas of Physics, from solitons and integrable field theories to Maxwell’s electrodynamics and non-abelian gauge theories describing the fundamental interactions of Nature.

Design of integrable quantum devices

The precise control of quantum systems will play a major role in the realization of atomtronic devices. Here we study models of dipolar bosons confined to three-well and four-well potentials. The analysis considers both integrable and non-integrable regimes within the models. Through variation of the external field, we demonstrate how the triple-well system can be controlled between various “switched-on” and “switched-off” configurations [1] and how the four-well system can be controlled to generate and encode a phase into a NOON state [2,3]. We also discuss the physical feasibility through use of ultracold dipolar atoms in BECs (three-wells) or optical superlattices (four-wells). Our proposals showcase the benefits of quantum integrable systems in the design of quantum devices.
[1] K. Wittmann, L. Ymai, A.Tonel, J. Links and A. Foerster, Communications Physics 1, 91 (2018)
[2] D. Grün, K. Wittmann, L. Ymai, J. Links and A. Foerster, Communications Physics 5, 36 (2022)
[3] D. Grün, L. Ymai, K. Wittmann, A. Tonel, A. Foerster and J. Links, Physical Review Letters 129, 020401 (2022)

Separating spin and charge in one dimensional atomic Fermi gas

Low-energy excitations of one dimensional (1D) interacting fermions typically split into two independent Tomonaga-Luttinger liquid (TLLs), each of which carries either spin or charge. This phenomenon is called spin-chargeseparation–a hallmark of 1D many-body physics. Although there have been more than 40 years of research in this field, theoretical understanding and definite experimental observation of this phenomenon are still challenging. In this talk I will discuss rigorous theoretical results of 1D spin-1/2 Fermi gas and our recent precise observation of the TLL theory of spin-charge separation in the trapped 1D ultracold Fermi gas. This work experimentally verifies the TLL theory of the spin-charge separation and provides strong evidence for nonlinear TLL effects, beyond the TLL model. It is a paradigmatic example of quantum many-body physics, offers new insight into quantum precision measurement.

Refs: Phys. Rev. Lett. 125, 190401 (2020), Science 376, 1305 (2022).

Frustrated Bearings

A bearing is a system of spheres (or disks) in contact. If in a bearing every loop must be even, one can obtain “bearing states”, in which touching spheres roll on each other without slip. We frustrate a system of touching spheres by imposing two different bearing states on opposite sides and search for the configurations of lowest energy dissipation. For Coulomb friction (with random friction coefficients) in two dimensions, a sharp line separates the two bearing states and we prove that this line corresponds to the minimum cut. Astonishingly however, in three dimensions, intermediate bearing domains, that are not synchronized with either side, are energetically more favourable than the minimum-cut surface. This novel state of minimum dissipation is characterized by a spanning network of slip-less contacts that reaches every sphere. Such a situation becomes possible because in three dimensions bearings of loops of size four have four degrees of freedom. By considering spheres of different size, packings with bearing states can even be made space-filling. The construction and mechanical properties of such space-filling bearings will be discussed. Space-filling bearing states can be viewed as a realization of solid turbulence exhibiting Kolmogorov scaling and anomalous heat conduction. Bearings states can be perceived as physical realizations of networks of oscillators with asymmetrically weighted couplings. These networks can exhibit optimal synchronization properties through tuning of the local interaction strength as a function of node degree or the inertia of their constituting rotor disks through a power-law mass-radius relation. As a consequence, one finds that space filling bearings synchronize fastest, when they are hollow.

Features of the Kitaev chain with long range pairing and hopping

We study variations of the Kitaev chain by including long-range pairing and hopping interactions. We discuss the phase diagrams resulting from the inclusion of long-range pairing first, then by introducing long-range hopping and finally by including both, pairing and hopping terms, using numerical tools in finite chains. Interestingly, some phases present Majorana zero-energy modes while others show massive edge modes. The absence of gap closing frontiers, namely, absence of truly phase transitions indicates aspects not present in the short-range well-known solution. We will discuss further on this topic. Issues of quantum entanglement will also be addressed.

Nonabelian Gauge Theories and Statistical Mechanics: a Lattice Affair

The nonlinear effects that characterize non-Abelian gauge theories such as QCD are responsible for the theory’s intriguing low-energy properties, and also for the great difficulty in investigating them analytically. The lattice formulation, introduced by K. Wilson in 1974, allows the study of theory by statistical mechanics methods, such as numerical (Monte Carlo) simulation, and provides the physical limit as a controlled extrapolation. In fact, lattice QCD constitutes a first principles approach, allowing the non-perturbative investigation of the theory’s low-energy properties, such as color confinement and the spontaneous breaking of chiral symmetry, which are associated with the generation of the mass of the visible universe. We will present and discuss results of this research area, describing general aspects of the formulation of gauge theories on the lattice and its numerical simulation.

Duality and Thermal Deformation in the Tricritical Ising Model

The thermal deformation of the 2D Tricritical Ising Model gives rise to an exact scattering theory with seven massive excitations based on the exceptional E₇ Lie algebra. The high and low temperature phases of this model are related by duality. We will discuss the peculiar features of the self-duality of this model and the possibility to compute exactly the matrix elements (Form Factors) of local and non-local magnetization operators.  We employ these Form Factors to compute the exact dynamical structure factors of the theory, a set of quantities with a rich spectroscopy which may be directly tested in future inelastic neutron or Raman scattering experiments.

Impurities in the extended Hofstadter-Hubbard model

The Hofstadter model is a lattice model to understand the Integer Quantum Hall effect. A lot is known about this model and its topological properties. Nevertheless, recently, it has been found that interactions can play an important role in its properties. A particular way to introduce interactions is to consider a Hubbard type interaction, thus defining what is known as the Hofstadter-Hubbard model. In this seminar I will discuss the influence of interactions in the dynamical behavior of local impurities in the pi-flux regime of the Hofstadter-Hubbard model.

Free fermionic quantum chains with inhomogeneous couplings

A new family of free fermionic quantum spin chains with multispin interactions was recently introduced. In this work, we argue that it is possible to build quantum Ising chains with inhomogeneous couplings with the same spectrum of the multispin chains. This is done by associating an antisymmetric tridiagonal matrix to the polynomials that characterize the quasienergies of the system via a modified Euclidean algorithm. The phase diagram of the inhomogeneous models is investigated numerically. It is characterized by gapped phases separated by critical hyperplanes with order-disorder transitions depending on the parity of the number of generators of the Hamiltonian, pointing out an underlying topological phenomena.

The partition function of the Sp(4) integrable vertex model

Numerics and  Heuristics  for the Lee-Yang partition function zeros of some antiferromagnetic Ising models

The well known Lee-Yang circle theorem provides a solution to the problem of locating the (grand) partition function zeros of the Ising ferromagnet. By adding the powerful technique of Asano contractions and the Trotter formula,  this method also covers the case of several quantum models. However these methods are not applicable to antiferromagnets, which remain poorly understood. In this talk I will discuss the antiferromagnetic models, partly using recent numerics  on the 2-d Ising model. I will also discuss another antiferromagnetic model analytically. This mean field type model is simpler than the original problem, but it is far from trivial. Our (largely) analytical solution provides useful insights into the original  problem.

Algebraic Bethe Circuits

The Algebraic Bethe Ansatz can be thought of as a tensor network to construct the eigenstates of several quantum many body systems like the XXZ model. This talk describes how to transform this tensor network into a quantum circuit and its physical realization in small quantum computers.

Solitons in Kitaev Spin Chains

The bond-dependent Ising interaction present in the Kitaev model has attracted considerable attention. The attention has mostly focused on the two-dimensional honeycomb lattice version of the Kitaev model but one can also imagine realizing simpler one-dimensional Kitaev spin chains that still has a surprisingly rich structure. It is usually assumed that for the Kitaev spin chains the presence of a magnetic field does not lead to any interesting new physics. However, recently we have found an unusual chiral soliton phase appearing when the field is applied in specific directions. The phase is centered around a special point where a two-fold degenerate ground-state can be exactly found. In this talk I will present some of the interesting physics related to the soliton phase for both integer and half-integer spin and discuss a simple variational picture of the soliton phase.

Integrable Reversible Cellular Automata

I will discuss a few examples of reversible, deterministic, discrete variable, and space-time discrete dynamical systems for which various dynamical quantities, such as space-time correlators, can be computed exactly. These models, athough interacting, often satisfy stronger algebraic constraints than Yang-Baxter equation, and may be considered superintegrable. The prime example is the so-called Rule 54 reversible cellular automaton, but I will discuss also a few other models as well as their quantum or stochastic deformations.

Sum of powers of principal minors

Sum of powers of principal minors (SPPM) of matrices appears in the calculation of many quantities in physics and applied mathematics. We show that the calculation of the Renyi entropy of fermionic systems, partition function of the Hubbard model and certain problems in machine learning are related to the SPPM problem. Although there is a simple formula to calculate the sum of principal minors of arbitrary matrices, it has been already shown that calculation of other powers is a hard problem (probably no polynomial time algorithm exists). In this talk I will first discuss how the Renyi (Shannon) entropy in quantum chains detects different phases and determines the universality classes. Then I will show how calculating this quantity boils down to a SPPM problem in certain quantum systems such as Ising chain and free fermions. Finally in the main part of the talk I will write a Grassmann representation for a generic matrix SPPM problem. This field theory-like representation shows interesting symmetries such as U^n(1),  symmetric group,  axial U(1), chiral and particle-hole symmetry. Using this representation one can make a mean-field approximation and find a reasonable estimate for the Renyi entropy which can detect the phase transition. We show that some of the mentioned symmetries are broken in, for example, the ferromagnetic phase of the Ising chain.